3.918 \(\int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx\)

Optimal. Leaf size=60 \[ \frac{d (c+d \sin (e+f x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right )}{a^2 f (n+1) (c-d)^2} \]

[Out]

(d*Hypergeometric2F1[2, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a^2*(c - d)
^2*f*(1 + n))

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Rubi [A]  time = 0.109314, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2833, 68} \[ \frac{d (c+d \sin (e+f x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right )}{a^2 f (n+1) (c-d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]

[Out]

(d*Hypergeometric2F1[2, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a^2*(c - d)
^2*f*(1 + n))

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{d x}{a}\right )^n}{(a+x)^2} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{d \, _2F_1\left (2,1+n;2+n;\frac{c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a^2 (c-d)^2 f (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0641619, size = 61, normalized size = 1.02 \[ \frac{d (c+d \sin (e+f x))^{n+1} \, _2F_1\left (2,n+1;n+2;-\frac{c+d \sin (e+f x)}{d-c}\right )}{a^2 f (n+1) (d-c)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]

[Out]

(d*Hypergeometric2F1[2, 1 + n, 2 + n, -((c + d*Sin[e + f*x])/(-c + d))]*(c + d*Sin[e + f*x])^(1 + n))/(a^2*(-c
+ d)^2*f*(1 + n))

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Maple [F]  time = 0.884, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( fx+e \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}}{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^2,x)

[Out]

int(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)/(a*sin(f*x + e) + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(-(d*sin(f*x + e) + c)^n*cos(f*x + e)/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(c+d*sin(f*x+e))**n/(a+a*sin(f*x+e))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)/(a*sin(f*x + e) + a)^2, x)